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Uniform negative immersions and the coherence of one-relator groups

Larsen Louder, Henry Wilton

Abstract

Previously, the authors proved that the presentation complex of a one-relator group $G$ satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of $G$ is free. Here, we prove that one-relator groups with negative immersions are coherent, answering a question of Baumslag in this case. Other strong constraints on the finitely generated subgroups also follow such as, for example, the co-Hopf property. The main new theorem strengthens negative immersions to uniform negative immersions, using a rationality theorem proved with linear-programming techniques.

Uniform negative immersions and the coherence of one-relator groups

Abstract

Previously, the authors proved that the presentation complex of a one-relator group satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of is free. Here, we prove that one-relator groups with negative immersions are coherent, answering a question of Baumslag in this case. Other strong constraints on the finitely generated subgroups also follow such as, for example, the co-Hopf property. The main new theorem strengthens negative immersions to uniform negative immersions, using a rationality theorem proved with linear-programming techniques.

Paper Structure

This paper contains 19 sections, 29 theorems, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs and pre-graphs
  4. Complexes and pre-complexes
  5. Asterisks and regular neighbourhoods
  6. Various kinds of maps
  7. Gluing pre-complexes
  8. Irreducible complexes
  9. Unfolding
  10. Irreducible cores
  11. Curvature conditions
  12. Wise's notion of negative immersions
  13. Stable commutator length
  14. A linear system
  15. Vertex and edge pieces
  16. ...and 4 more sections

Key Result

Theorem A

Let $G=F/\mathord{\langle} \mathord{\langle}w\mathord{\rangle} \mathord{\rangle}$ be a one-relator group. If every two-generator, one-relator subgroup of $G$ is free, then $G$ is coherent.

Figures (3)

  • Figure 1: A pre-complex $(G,S,w)$: $G$ is a theta graph with two half-edges sticking off, and $S$ is the union of two open arcs and one cycle. The map $w$ is a closed immersion, since it maps $\partial S$ to $\partial G$.
  • Figure 2: The central diagram illustrates the complex $X$ from Example \ref{['eg: Folding a torus']}: the 1-skeleton is coloured red, and the single face is attached along the black curve. It admits two different unfoldings, each a realisation of a torus.
  • Figure 3: An example of a vertex piece.

Theorems & Definitions (104)

  • Definition 1.1: Coherence
  • Theorem A
  • Theorem 1.2: louder-wilton2
  • Remark 1.3
  • Theorem B
  • Remark 1.4
  • Theorem C
  • Conjecture 1.5
  • Example 1.6
  • Conjecture 1.7: Heuer's conjecture
  • ...and 94 more